What is Minimum: Definition and 1000 Discussions

In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given range (the local or relative extrema), or on the entire domain (the global or absolute extrema). Pierre de Fermat was one of the first mathematicians to propose a general technique, adequality, for finding the maxima and minima of functions.
As defined in set theory, the maximum and minimum of a set are the greatest and least elements in the set, respectively. Unbounded infinite sets, such as the set of real numbers, have no minimum or maximum.

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  1. S

    Find minimum value of f(x) in terms of variable a

    (1) For ##x>a## ##f(x)=x^2+x-a+1 \rightarrow## minimum value obtained when ##x=-\frac{1}{2}## Minimum value of ##f(x)=\frac{3}{4} -a## (2) For ##x<a## ##f(x)=x^2-x+a+1 \rightarrow## minimum value obtained when ##x=\frac{1}{2}## Minimum value of ##f(x)=\frac{3}{4}+a## But the teacher said...
  2. Baums Mizushala

    Point on a graph nearest to the origin

    The Attempt at a Solution I know the answer is supposed to be ##(-1,0)##. However when I differentiate the above expression I get. $$ 2x+{\frac 5 2} $$ Then the shortest distance would be when the expression equates to 0. $$ 2x+{\frac 5 2}=0 $$ I should be getting ##x=-1## but solving for ##x##...
  3. A

    Determining the minimum velocity when given height and length

    Now I've tried looking at the problem like this. Considering that a is the length off the vehicles that he is trying to jump over I would consider that to be s. The plane from which he starts (b) should be the h. So considering that he is jumping from a horizontal plane, gravity should also...
  4. anemone

    MHB Min Value of $\dfrac{a+3c}{a+2b+c}$+$\dfrac{4b}{a+b+2c}$+$\dfrac{8c}{a+b+3c}$

    Let $a,\,b$ and $c$ be positive real numbers. Determine the minimum value of $\dfrac{a+3c}{a+2b+c}+\dfrac{4b}{a+b+2c}+\dfrac{8c}{a+b+3c}$.
  5. A

    Minimum value of an angle

    I tried to do it by derivative but there are two variables, so I don't know how to proceed. Does anyone know how I can solve it? Remembering that you don't need to find the value of ##\theta##. I just need to find a relationship between ##\theta_1## and ##\theta_2##
  6. derpydashie6167

    Minimum work needed to climb to the top of a peak

    f=ma=71.5*9.8 = 700.7 I know this is not right because he is also going up against gravity but I don't know what else to use for acceleration. I don't know the angle but I assume it is a 90 degree cliff. w = (Fcos90)1380 = 0. But zero is not the correct answer.
  7. anemone

    MHB Maximum and minimum of a function

    Let $a$ be an integer. Consider the function $y=\dfrac{12x^2-12ax}{x^2+36}$. For what integral values of $a$ the maximum and the minimum of the function $y=f(x)$ are integers?
  8. S

    Mechanics: Find the minimum initial velocity and angle of this projectile

    Attempt at solution in attachment.
  9. jaychay

    MHB Determine the relative maximum and minimum on the graph

    Given that f is the function on (−∞, ∞) and the graph is the derivative of f 1.) Find the critical point on the graph ? 2.) Find the interval of the increasing function on the graph ? 3.) Find the interval of the decreasing function on the graph ? 4.) Find the point which is the absolute...
  10. G

    MHB Finding minimum value of function with two variable

    I have a formula for cost calculation that contain x and y two variable. I have to find the value of (x,y) where that formula will gives minimum value as cost should not be equal to zero, it has some minimum value. I took 1st partial derivative with respect to x and then with y and found the...
  11. tanaygupta2000

    Minimum average value of position

    After getting the values of ψ₀(x) and ψ₁(x), I put them in the expression of ϕ(x) to get: ϕ(x) = (mw/πℏ)^(1/4) * exp[-(mw/2ℏ)x^2] * [α + βx√(2mw/ℏ)] Now when attempting to find the value of <x> by ∫xϕ(x) dx, I am having trouble determining the limits, as I am getting nothing useful by...
  12. P

    Minimum force required to rotate a lamina

    When the lamina rotates about A, FA must act on B (because it is the farthest away) perpendicular to AB (so that all of FA contributes to rotation). Same argument is valid for rotation of lamina about B as well. Having noted that, I tried two approaches: Approach 1- If I assume that the...
  13. E

    Minimum distance between balls connected by rods

    I defined the angle ##\beta## as the angle from the right horizontal to the ball C, from B, and ##\alpha## as the angle from the left horizontal to the ball A, from B. I also work in the CM frame, which has a velocity downwards of magnitude ##\frac{v}{3}## w.r.t. the lab frame. The positions of...
  14. Kaguro

    Minimum time between two orthogonal states

    E = (1/√2)^2(E1) + (1/√2)^2(E2) = (E1+E2)/2 Let ψ(x,t=0) = ψ0 So, ψ1 = ψ0*exp(-i*E*T1/ħ) and, ψ2 = ψ0*exp(-i*E*T2/ħ) Given, <ψ1|ψ0> = <ψ2|ψ0> = 0 So, <ψ0*exp(-i*E*T1/ħ)|ψ0> = 0 => exp(i*E*T1/ħ)<ψ0|ψ0> = 0 => exp(i*E*T1/ħ) = 0 Similarly, exp(i*E*T2/ħ) = 0 So, exp(i*E*T1/ħ) = exp(i*E*T2/ħ)...
  15. CricK0es

    Minimum number of numbers to express every integer below N as a sum

    I have found code to find simply the minimum numbers needed, but I need to do it without repetition given the nature of an electric circuit. I hope that is a sufficient enough explanation of the problem. Despite being an engineering project this aspect is more mathematical.
  16. LCSphysicist

    Prisoner sliding down a rope with minimum velocity

    I thought in this equations f is the man's pull\ f + dm*g = T < 600 Where dm is equal to the mass of the string that pull the up part (15-x) after descending x meters. dm/(15-x) = m/15 And, to the man: W - f = Mx'' I can solve this, and i got ~8m/s Is this right?
  17. L

    Train deceleration and minimum stopping distances

    So my work includes using the acceleration formula a=delta v/t (Vrtf-Vrti)/a -> (0-54)/(-0.31) -> t=174 seconds I plug in 174 seconds to find the acceleration of the left train. and got -0.22m/s^2 I then used the displacement equation x=(1/2)at^2+Vo+So coming out with Xrt=4703m and Xlt=...
  18. I

    I Why is 2.5 times the radius the minimum height needed to do a loop?

    The height can be determined by conservation of energy (ignoring all friction). The mechanical energy when the car is at rest, equals the mechanical energy when the car is in the middle of the loop (at the top of the loop): \begin{equation} E_{0} = E_{loop} \\ mgh_0 = \frac{1}{2}mv^2+mgh_{loop}...
  19. G

    MHB Quadratic equations intersaction point is minimum instead of roots

    I have 2 quadratic functions and I am interested in their root in the specific range. I use quadratic equation to get their roots and what I find that if their any real solution exist for both or any of the function that lie in it designated specific range, then the roots are maximum or minimum...
  20. karush

    MHB -b.2.2.26 Solve first order IVP and determine where minimum of solution occurs

    OK going to comtinue with these till I have more confidence with it $$\dfrac{dy}{dx}=2 (1+x) (1+y^2), \qquad y(0)=0$$ separate $$(1+y^2)\, dy=(2+2x)\, dx$$
  21. LCSphysicist

    What is the minimum mathematic requirement for learning Lagrangian and Hamiltonian mechanics?

    Homework Statement:: ... Relevant Equations:: . What is the minimum mathematic requirement to the Lagrangian and hamiltonian mechanics? Maybe calc 3 and linear algebra?
  22. I

    Question about calculating the minimum temperature in hot air balloons

    First, I tried using the Archimedes principle and calculated the weight of the surrounding air displaced when taking off. ##W = 2500\times 1.29\times 9.81 = 31637.25 N## But then, I got stuck and do not know how to proceed from here on. I don't want the full solution yet but can I get some...
  23. karush

    MHB S8.3.7.6 minimum vertical distance

    S8.3.7.6. What is the minimum vertical distance between the parabolas $$y = x^2+1 \textit{and } y = x- x^2$$ Ok I think what question is ... The vertical distance between vertex's
  24. karush

    MHB S8.3.7.3. whose sum is a minimum

    S8.3.7.3. Find two positive numbers whose product is 100 and whose sum is a minimum $x(100-x)=100x-x^2=100$ So far Looks like it's 10+10=20Doing all my lockdown homework here since I have no access to WiFi and a PC. and just a tablet where overkeaf does not work
  25. U

    Understanding the Minimum Speed to Keep Carriage on Tracks in a Loop

    I recognise that the normal force must alwayss act towards the centre of the circle loop, as the rail always has to be exertign a pushing force on the car/carriage in order for it to follow the trajectoryof the loop. However , I cannot understand why, the reaction force has to be greater than...
  26. J

    MHB Minimum Degree of a Random Graph (Probabilistic Method)

    Problem: Suppose that the function $p : N \rightarrow [0, 1]$ satisfies $p >> n^{-1}ln(n)$ (i.e. $n^{-1}ln(n) = o(p)$). (a) Prove that as $n \rightarrow \infty$, the random graph $G(n, p)$ has minimum degree at least $\frac{np}{2}$ almost surely. Idea: Look at the degree of each individual...
  27. E

    Minimum frequency for a point to have maximum amplitude in standing wave

    When I tried using the equations the only thing I could see is that it is impossible for such point to be an anti-node. In this case, how do I find the frequency? The answer is not even with the form of v*n/2L which is very confusing to me, I thought that the frequency of a standing wave must...
  28. AngelFis93

    Minimum wavelength of phonons under the Debye aproximation

    Since in Debye aproximation Debye's frecuency is defined as the maximum frecueny , the corresponding wavelenght should be the minimum one, due to the inverse relation among those λ=v/f=v·2π/ω=5.9 Å , which is higher than the given result. I believe I should be using the information 'cubic...
  29. L

    Why Do Physical Systems Seek Minimum Potential Energy?

    Many, many years ago while in engineering graduate school I was studying calculus of variations. One classic problem was to determine the shape of a hanging cable supported at its two ends. After minimizing the integral, the catenary curve was the solution. The basic assumption in setting up...
  30. sergey_le

    Exploring Local Minimum and Maximum Points in Continuous Functions

    Here's what I tried to do: f Continuous function at R, x1 local minimum point of f, x2 local maximum point of f. Existing f(x1)>f(x2). Let's look at the interval [x1,x2]⊆ℝ . f is continuous in R and therefore continuous in its partial segment. Therefore f continuous in [x1,x2]. Therefore, there...
  31. Uchida

    I Minimum number of cycles in a short pulse laser

    Hello to all, In a short pulse laser emission setup, can a pulse be emmited with beam length shorter than one wavelenght? (can a pulse have a duration shorter than its period?) Lets say a laser emmiter shoots a quarter cycle pulse, what would happen to this short beam? (lets supose the...
  32. nineteen

    What is the minimum velocity needed?

    I tried to solve this problem and this is what I could come through: When the object is moving, the force acting on object is the frictional force, so, it got to be μmg. So, F = ma and as F is μmg μmg = ma μg = a So, to find out the magnitude of the initial velocity v given to the smaller...
  33. G

    MHB Finding the Minimum Number of White Balls in a Container with 27 Balls

    We have 27 balls in the container, some of which are white and some black. How many white balls in the container must be at least, so that the probability that two black balls were drawn at random without a return was less than 23/30?
  34. C

    B How Can We Prove That x^x Reaches a Minimum at x = e^-1?

    I've noticed that x^x is a minimum for x = e^-1 I put it as a high school problem because I presume it's one of those simple differential proofs/identities, but I can't really see how to get to e^-1. Too long since I did any calculus. Can someone please show me how to arrive at that? How about...
  35. pkress

    Engineering Minimum Critical Power Ratio (nuclear engineering applications)

    Im having a really tough time with this problem, I am assuming that in order for q'(z) to be a maximum, e^(-az/L)sin (pi(z)/L) must be a maximum. I believe this occurs when the derivative with respect to with respect to z/L is zero, which gives me z = 0.322L, but I am not sure if this is correct...
  36. zizzle

    Minimum initial speed to spin a particle around a disk (with gravity)

    For this problem, since the weight force on the "particle" (child) is not always aligned with the tangential circular path of the disks, I couldn't think of a way to use rotational kinematics equations. As such, I tried to solve the problem using work principles (namely, that the change in...
  37. Monoxdifly

    MHB [ASK] Minimum Dimension of a Map

    The farthest distance of two places in an area is 200 km. If someone wants to make a map of that area on a 1 m × 1 m paper, the possible scale to make it is ... a. 1 : 210 b. 1 : 2.100 c. 1 : 21.000 d. 1 : 210.000 Can you help? The 200 and 210 makes me think that the distance on map won't be an...
  38. S

    I Find the minimum and maximum value of a quadratic form

    By working with the following definition of minimum of a quadratic form ##r(\textbf{x})##, ##\lambda_1=\underset{||\textbf{x}||=1}{\text{min}} \ r(\textbf{x})## where ##\lambda_1## denotes the smallest eigenvalue of ##r##, how would one tackle the above problem? It is clear that the diagonal...
  39. U

    Understanding the Math Behind A: Minimum Length of Plane Mirror

    15. What is the minimum length of a plane mirror in order for you to see a full view of yourself? A 1/2 your height B 1/4 your height C 3/4 your height D your full height Q Why is answer of A given is the correct one, I understand pictorially how it is, since visually if you were to draw a...
  40. K

    Minimum work required to move a point charge

    This is the figure for the problem: 1.) Solved for initial total EPE of the system EPE system = (kq2q3/a) + (kq2q1/b) + (kq1q3/√a^2 + b^2) 2.) Solved for final EPE of the system negating q1 as if it were off to infinity EPE system final = (kq2q3/a) 3.) Plugged values into equation W =...
  41. I

    Minimum separation between incoming proton and alpha particle

    Proton is going towards the ##\alpha## particle. So, I am thinking of using the conservation of energy as the initial kinetic energy of the proton is known and initial interaction potential energy is zero. But, we don't know the kinetic energies of proton and ##\alpha## particle when they are at...
  42. K

    Minimum distance between two perpendicular moving vehicles

    Below is my solution, which I'm not sure is correct. Hopefully the picture shows clearly what I'm trying to find.
  43. Yalanhar

    Minimum velocity to throw a object to another planet

    So I integrated the work done on the object by both planets. Work1 is until x, and Work2 is from x to d. Where x is the point where both gravitational forces are equal. ##W_1=\int_0^x \frac{GMm}{r^2}dr - \int_0^x \frac{GMm}{(3R+D-r)^2}dr ## ##W_2=\int_x^D \frac{GMm}{(3R+D-r)^2}dr - \int_x^D...
  44. A

    I Interacting fields in QFT and minimum excitations....

    Following up on @A. Neumaier's excellent series of articles on virtual particles, I'm confused about one thing (well of of several). If you pop over to the discussion of virtual particles on Matt Strassler's page, he mentions that, for example, an excitation in the photon (em) field will also...
  45. G

    Calculation of minimum angular velocity of a mass on a spinning plate

    Problem Statement: How to calculate minumum angular velocity of a mass on a spinning plate Relevant Equations: f=mrw^2 Hi, here's the question: a) A rough horizontal plate rotates with a constant angular velocity of w about a fixed vertical axis. A particle of mass m lies on the plate at a...
  46. dRic2

    Finding the minimum of an integral with Lagrange multipliers

    Using Lagrange multiplier ##\lambda## (only one is needed) the integral to minimize becomes $$\int_{\tau_1}^{\tau_2} (y + \lambda) \sqrt{{x'}^2+{y'}^2} d \tau = \int_{\tau_1}^{\tau_2} F(x, x', y, y', \lambda, \tau) d\tau $$ Using E-L equations: $$\frac {\partial F}{\partial x} - \frac d {d \tau}...
  47. P

    Minimum force required to prevent sliding down

    Recently I've come across a question that seems very simple, but had puzzled me for a while. Suppose a point object with mass M is placed on a rough plane inclined at 30 degree to the horizontal and is subjected to the force of gravity acting down vertically (to make it simple, assume g = 10...
  48. Efeguleroglu

    How to calculate the minimum kinetic energy (special relativity)?

    Sadly, that's what all I could do.
  49. T

    What is the minimum force required to move the box?

    Problem Statement: What is the minim Force F necessary to make the crate start moving up the incline? Relevant Equations: F_push=mgsin(ø)+F_f F_f= µ_s mgcos(ø) My values m = 80kg ø = 20 Fø = 15 static friction = 0,5 constant friction = 0,4 F _push = 80kg * 9.8 m/s^2 *sin(20) + 0,5 * 80kg *...
  50. E

    Comp Sci Using C++ to find the minimum number of something

    I attempted going with c++ and this is what i got: #include <iostream> Using namespace std; int i, j, keeper, n; int Ingredients[100], Blender[100], cout<<”How many ingredients does the shake need?”<<endl; cin>>n; cout<<”Type down the ingredients:”<<endl; for(i=1; i<=n; i++) {...
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